Abstract
We show that use of ordinary least-squares to explore relationships involving firm-level stock returns as the dependent variable in the face of structured dependence between individual firms leads to an endogeneity problem. This in turn leads to biased and inconsistent least-squares estimates. A maximum likelihood estimation procedure that will produce consistent estimates in these situations is illustrated. This is done using methods that have been developed to deal with spatial dependence between regional data observations, which can be applied to situations involving firm-level observations that exhibit a structure of dependence. In addition, we show how to correctly interpret maximum likelihood parameter estimates from these models in the context of firm-level dependence, and provide a Monte Carlo as well as applied illustration of the magnitude of bias that can arise.
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Notes
In the trivial case where ρ = 0, ϕ = 0, so there is no significant dependence least-squares would be unbiased. This would also be the case if the matrices W,V were both strictly triangular. However, triangularity of these matrices is not possible because our sample of firms cannot be sorted by both metropolitan location and industry membership in such a way as to produce W,V that are both triangular.
In substantive applications one would want to determine an ‘optimal’ number of neighboring firms to employ using model comparison criterion (see LeSage and Pace 2009, Chapter 6). For our illustrative purposes here this is not an important issue, and in fact the matrix W L contains some non-zero blocks on the diagonal for cases where the 30 nearest neighboring firms all lie within the same county. This prevents spatial spillovers to neighboring counties from arising in the cross-partial derivatives unless there are firms in the same industry located in nearby counties.
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Moon, K.P., LeSage, J.P. Simultaneous dependence between firm-level stock returns. J Econ Finan 37, 479–494 (2013). https://doi.org/10.1007/s12197-011-9188-5
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DOI: https://doi.org/10.1007/s12197-011-9188-5